In a small bowl, combine the ginger, garlic, soy sauce, vinegar, sriracha, and sesame oil.
Heat the neutral oil in a large skillet over medium heat. Add the chicken in one piece and cook, undisturbed, until well browned on the bottom, 2-3 minutes. Flip and brown the other side, breaking the chicken up into 3-4 large pieces. Once the chicken is well browned on both sides, start stirring to break it up into small pieces.
Add the light parts of the scallions, carrots, and cabbage, and stir well. Add the sauce mixture and stir to combine until the chicken is no longer pink and the sauce is evenly distributed. Sprinkle with sesame seeds, the greens of the scallions, and cilantro. Serve in lettuce cups with rice if you like.
Store the filling for the chicken lettuce wraps in an airtight container in the fridge for 3 days, or in the freezer for 3 months, then add to the lettuce when you're ready to serve.